On summability of N-fold Fourier integrals correspondingto pseudodifferential operators.
It is well known, that if N ≥ 3, then spherical partial sums of N-fold Fourier integrals (eigenfunction expansions of Laplace operator) of the characteristic function of the unit ball diverge at the origin. Note, here level surface of Laplace operator and the surface of discontinuity of the consider...
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| Format: | Article |
| Language: | English |
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2010
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| Online Access: | http://psasir.upm.edu.my/id/eprint/23265/ http://psasir.upm.edu.my/id/eprint/23265/1/On%20summability%20of%20N.pdf |
| _version_ | 1848844709147967488 |
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| author | Ahmedov, Anvarjon Ashurov, Ravshan |
| author_facet | Ahmedov, Anvarjon Ashurov, Ravshan |
| author_sort | Ahmedov, Anvarjon |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | It is well known, that if N ≥ 3, then spherical partial sums of N-fold Fourier integrals (eigenfunction expansions of Laplace operator) of the characteristic function of the unit ball diverge at the origin. Note, here level surface of Laplace operator and the surface of discontinuity of the considered piecewise smooth function are both spheres. It was first noted by Pinsky and Taylor, that if we consider nonspherical partial sums (eigenfunction expansions of elliptic pseudodifferential operators), then to obtain the same effect, we should change the above second sphere with the dual set to level surface of the pseudodifferential operator. Namely, nonspherical partial sums of a piecewise smooth function, supported inside the dual surface converge everywhere except the origin. In this paper we investigate summability of these expansions by Riesz method and show that the order s > (N − 3)/2 of Riesz means guarantees convergence everywhere, where the function is smooth. Since piecewise smooth functions are in Nikolskii class H 1 1 (R N ), we also establish necessary and sufficient conditions for uniform convergence of expansions of H p a -functions. |
| first_indexed | 2025-11-15T08:35:14Z |
| format | Article |
| id | upm-23265 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T08:35:14Z |
| publishDate | 2010 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-232652015-11-03T03:14:11Z http://psasir.upm.edu.my/id/eprint/23265/ On summability of N-fold Fourier integrals correspondingto pseudodifferential operators. Ahmedov, Anvarjon Ashurov, Ravshan It is well known, that if N ≥ 3, then spherical partial sums of N-fold Fourier integrals (eigenfunction expansions of Laplace operator) of the characteristic function of the unit ball diverge at the origin. Note, here level surface of Laplace operator and the surface of discontinuity of the considered piecewise smooth function are both spheres. It was first noted by Pinsky and Taylor, that if we consider nonspherical partial sums (eigenfunction expansions of elliptic pseudodifferential operators), then to obtain the same effect, we should change the above second sphere with the dual set to level surface of the pseudodifferential operator. Namely, nonspherical partial sums of a piecewise smooth function, supported inside the dual surface converge everywhere except the origin. In this paper we investigate summability of these expansions by Riesz method and show that the order s > (N − 3)/2 of Riesz means guarantees convergence everywhere, where the function is smooth. Since piecewise smooth functions are in Nikolskii class H 1 1 (R N ), we also establish necessary and sufficient conditions for uniform convergence of expansions of H p a -functions. 2010 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/23265/1/On%20summability%20of%20N.pdf Ahmedov, Anvarjon and Ashurov, Ravshan (2010) On summability of N-fold Fourier integrals correspondingto pseudodifferential operators. Journal of Pseudo-Differential Operators and Applications, 1 (1). pp. 417-432. ISSN 1662-9981 10.1007/s11868-010-0017-y |
| spellingShingle | Ahmedov, Anvarjon Ashurov, Ravshan On summability of N-fold Fourier integrals correspondingto pseudodifferential operators. |
| title | On summability of N-fold Fourier integrals correspondingto pseudodifferential operators. |
| title_full | On summability of N-fold Fourier integrals correspondingto pseudodifferential operators. |
| title_fullStr | On summability of N-fold Fourier integrals correspondingto pseudodifferential operators. |
| title_full_unstemmed | On summability of N-fold Fourier integrals correspondingto pseudodifferential operators. |
| title_short | On summability of N-fold Fourier integrals correspondingto pseudodifferential operators. |
| title_sort | on summability of n-fold fourier integrals correspondingto pseudodifferential operators. |
| url | http://psasir.upm.edu.my/id/eprint/23265/ http://psasir.upm.edu.my/id/eprint/23265/ http://psasir.upm.edu.my/id/eprint/23265/1/On%20summability%20of%20N.pdf |